Closed Form Solution for the Surface Area, the Capacitance and The

Home , Ellipsoid

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There is also a practical interest for an exact solution for the ellipsoidal surface area in various fields of science, we just mention a few such fields: 1) in biology the human cornea as well as the chicken erythrocytes are realistically described by an ellipsoid and the area is important in the latter case for the determination of the permeabilities of the cells [3],[4] 2) in cosmology and the physics of rotating black holes [5]and 3) in the geometry of hard ellipsoidal molecules and their virial coefficients. In particular in the latter case, the surface area appears in the expression for the pressure of the ellipsoidal molecules [6]. We also mention the relevance of the surface area of ellipsoid for the investigation and measurement of capillary forces between sediment particles and an air-water interface [7]. For an application to medicine we refer the reader to [8]. On the other hand there are two further important aspects related to the geometry of the ellipsoid awaiting for a full analytic solution with many important applications. Namely: first the calculation in closed analytic form of the capacitance of a conducting ellipsoid and second the exact analytic calculation of the demagnetizing factors of a magnetized ellipsoid. In the former case, the geometry of the ellipsoid is complex enough to serve as a promising avenue for modeling arbitrarily shaped conducting bodies [9]. Capacitance modulation has been suggested recently as a method of detecting microorganisms such as the E. coli present in the water [10]. Despite its importance in theory and applications, no exact analytic solution for the capacitance of the ellipsoid had been derived by previous authors. There was only a formula in terms of formal integrals derived in [9]. In the later case, the magnetic susceptibility κ of the body determined in the ambient magnetic field B~ is influenced by the shape and dimensions of the body. Thus the measured (apparent) magnetic susceptibility κA should be corrected for this shape effect to obtain the shape-independent true susceptibility κT . The relation between the true and apparent volume susceptibility involves the so called demagnetizing factors. The first attempts of calculating the demagnetizing factors of the ellipsoid were made in [19],[20]. However, the authors of these works only derived expressions in terms of formal integrals. In this paper, we derive for the first time the closed form solution for the three demagnetizing factors for the ellipsoid, in terms of the first hypergeometric function of Appell of two variables. A fundamental application of our work will be in the determination of asteroidal magnetic susceptibility and its comparison to those of meteorites in order to establish a meteorite-asteroid match [12]. An- other interesting application of our solution for the demagnetizing factors of the ellipsoid would be in the field of microrobots. An external magnetic field can induce torque on a ferromagnetic body. Thus the use of external magnetic fields has strong advantages in microrobotics and biomedicine such as wireless controllability and safe use in clinical applications [11]. Thus, there is a certain demand from pure and applied mathematics for the closed form solutions of the above geometric problems. It is the purpose of our paper to produce such novel and useful exact analytic solutions for all three described problems above. We report our findings in what follows.

2 2 Closed form solution for the surface area of the ellipsoid.

We consider an ellipsoid centred at the coordinate origin, with rectangular Cartesian coordinate axes along the semi-axes a,b,c :

x2 y2 z2 + + =1. (1) a2 b2 c2 We begin our exact analytic calculation for the inﬁnitesimal surface area dS, using the formula for the surface Monge segment :

~x(x, y) = (x,y,z(x, y)), (2)

dS = ~x ~x dxdy = 2 1+ z2 + z2dxdy, (3) | x × y| x y ∂z(x.y) ∂z(x,y) q where zx := ∂x , zy := ∂y and,

2 2 2 2 2 2 2 c 1 2 c 1 1+ zx + zy =1+ x 2 2 + y 2 2 a2 2 x y b2 2 x y c 1 2 2 c 1 2 2 − a − b − a − b c2 1 c2 1 =1+ x2 + y2 a4 x2 y2 b4 x2 y2 1 2 2 1 2 2 − a − b − a − b 2 2 2 2 2 2 2 2 2 2 c x c y x y x c y c 1 1 2 2 1 2 2 1 a2 b2 + a4 + b4 − − a a − − b b = − − 2 = 2 = x2 y x2 y 1 2 2 1 2 2 ⇒ − a − b − a − b (4)

Substituting to (3) we get

c2 x2 c2 y2 2 1 1 a2 a2 1 b2 b2 dS = − − − 2 − dxdy v x2 y u 1 a2 b2 u − − t x2 y2 2 1 δ a2 ε b2 = − − 2 dxdy, (5) v x2 y u 1 a2 b2 u − − t where we deﬁne: δ := 1 c2/a2, ε := 1 c2/b2. (6) − −

3 Consequently the octant surface area is given by:

2 a b √1 x2/a2 x2 y2 ellipsoid − 2 1 δ a2 ε b2 = − − 2 dy dx Aoct.  v x2 y  0 0 u 1 a2 b2 Z Z u − −  t 2  x  ε 2 2 2 2 1 δ 2 1 2 y a b √1 x /a a 2 x − − b 1 δ 2 − v − a =  u dy dx 2 0  0 u x 1 1 2   2 2 2  Z Z u 1 a 1 b x y  u − − 1 2 u − a 2 2 2 t a  b √1 x /a   − (1 µ2y2)  =  Ω − dy dx,  (7) (1 λ2y2 Z0 (Z0 s − ) with x2 1 δ 2 ε 1 1 2 − a 2 2 Ω := x2 , µ := 2 2 2 , λ := 2 x2 . (8) s 1 2 b (1 δx /a ) b 1 2 − a − − a We deﬁne a new variable:

y dy 2 2 2 y1 := = dy1 = , with η := b 1 x /a . (9) b 2 1 x2/a2 ⇒ η − − p Thus: p

a 1 1 µ 2y2 ellipsoid = Ω(x) − ′ 1 dy η dx Aoct. 1 λ 2y2 1 Z0 (Z0 s − ′ 1 ) a 1 2 1 µ′ ψ dψ = Ω(x)η(x) − 2 2 dx = (10) 0 0 s 1 λ′ ψ 2√ψ ⇒ Z (Z − )

a ellipsoid 1 Γ(1/2)Γ(1) 1 1 1 3 2 2 oκτ. = Ω(x)η(x) F1 , , , ,µ′ , λ′ dx, A 0 2 Γ(3/2) 2 −2 2 2 Z (11) with 2 2 2 2 2 (1 x /a ) 2 2 µ′ := η µ = − (1 c /b ), (12) (1 δx2/a2) − − 2 2 2 2 2 1 b (1 x /a ) λ′ := η = − = 1 (13) b2(1 x2/a2) b2(1 x2/a2) − − and F1(α,β,β′,γ,x,y) denotes the ﬁrst generalized hypergeometric function of Appell [13] with two variables x, y and parameters α,β,β′,γ :

∞ ∞ (α, m + n)(β,m)(β′,n) m n F (α,β,β′,γ,x,y)= x y . 1 (γ,m + n)(1,m)(1,n) m=0 n=0 X X (14)

4 The double series converges absolutely for x < 1, y < 1. Thus we obtain : | | | | a 2 ellipsoid 2 δx 1 Γ(1/2)Γ(1) 1 1 1 3 2 = b 1 F , , , ,µ′ , 1 dx Aoct. − a2 2 Γ(3/2) 1 2 −2 2 2 Z0 r a 2 2 δx 1 Γ(1/2)Γ(1) Γ(3/2)Γ(1/2) 1 1 2 = b 1 F , , 1,µ′ dx − a2 2 Γ(3/2) Γ(1)Γ(1) 2 −2 Z0 r a 2 2 2 2 δx π 1 1 (1 x /a ) = b 1 F , , 1, − (1 c2/b2) dx. − a2 2 2 −2 (1 δx2/a2) − Z0 r − (15) In the transition from the ﬁrst to the second line of the previous equation we made use of the property of Appell’s hypergeometric function according to which if one of its two variables is set to the value 1 (one), then the function F1 reduces to the ordinary hypergeometric function of Gauß:

1 1 1 3 2 Γ(3/2)Γ(3/2 1/2 1/2) 1 1 3 1 2 F , , , ,µ′ , 1 = − − F , , ,µ′ . 1 2 −2 2 2 Γ(3/2 1/2)Γ(3/2 1/2) 2 −2 2 − 2 − − (16) It is also valid: 1 2 ellipsoid 2 2 π 1 1 (1 x1) 2 2 oct. = ab 1 δx1 F , , 1, − 2 (1 c /b ) dx1. (17) A 0 − 2 2 −2 (1 δx ) − Z q − 1 We now apply the transformation: 1 x2 − 1 =1 η2 (18) 1 δx2 − − 1 which yields: 1 ellipsoid ab 1 π 1 1 2 = 2 F , , 1, (1 η )ε dη oct. 2 2 A √1 δ 0 δη 2 2 −2 − − Z 1+ 1 δ − 1 h i π/2 ab 2 1 = 1 εη2 dφdη 2 2 2 √1 δ 0 − 0 δ(1 η ) 2 − Z Z 1+ 1−δ cos φ p − 2 h δ(1 η ) i 1 − ab 2 π 2+ 1 δ 2 − = 2 1 εη 3/2 dη = (19) √1 δ 0 − 4 (δ(1 η2)) ⇒ − Z 1+ 1−δ p − the total surface area is given by h i 1 ellipsoid 8ab 2 2 π 1 = 2 1 εη 3/2 dη A √1 δ ( 0 − 2 δ(1 η2) − Z 1+ 1−δ p − n2 h io 1 π δ(1 η2) 1 εη2 + − − − dη , (20) 2 3/2 0 − 4 1 δ pδ(1 η ) ) Z − 1+ 1−δ − n h io

5 while using

δ(1 η2) 1 δ + δ(1 η2) 1 1+ − = − − = (1 δη2), (21) 1 δ 1 δ 1 δ − − − − we obtain:

ellipsoid 8ab π Γ(1/2)Γ(1) 1 1 3 = F1 ,βǫ, ,ε,δ A √2 1 δ 4 Γ(3/2) [1/(1 δ)]3/2 2 2 − ( − π ( δ) 1 1 Γ(1/2)Γ(2) 1 5 − F ,β , ,ε,δ − 4 1 δ [1/(1 δ)]3/2 2 Γ(5/2) 1 2 ǫ 2 − − ) (22) and we deﬁned the 2-tuple:

1 3 β := , . (23) ǫ −2 2 Equation (22) is our solution in closed analytic form for the surface area of the ellipsoid. We believe it constitutes the ﬁrst complete exact analytic solution of the problem, while equation (22) is of certain mathematical beauty. Thus, we have proved the theorem:

Theorem 1 The surface area of the general ellipsoid in closed analytic form is given by the equation:

ellipsoid 8ab π 1 1 3 π ( δ) 1 1 5 = 2 3 2 F ,β , ,ε,δ − 3 2 F ,β , ,ε,δ scalene √1 δ 2 [1/(1 δ)] / 1 2 ǫ 2 6 1 δ [1/(1 δ)] / 1 2 ǫ 2 A − ( − − − − ) ⇐⇒ 2 2 ellipsoid c 1 1 3 3 1 c 1 1 3 5 =4πab 2 F ( , , , ; ǫ,δ)+ 1 2 F ( , , , ; ǫ,δ) Ascalene a 1 2 − 2 2 2 3 − a 1 2 − 2 2 2 (24)

The two-variables function F1 (α,β,β′,γ,x,y) , admits the following integral representation which is of vital importance in the proof of the theorem eqn. (22), for the surface area of the general ellipsoid :

1 ′ α 1 γ α 1 β β Γ(α)Γ(γ α) u − (1 u) − − (1 ux)− (1 uy)− du = − F (α,β,β′,γ,x,y) − − − Γ(γ) 1 Z0 (25) We point out that in proving the theorem 1 we also produced the following interesting result:

6 Theorem 2 1 2 2 2 π 1 1 (1 x1) 2 2 1 δx1 F , , 1, − 2 (1 c /b ) dx1 0 − 2 2 −2 (1 δx ) − Z q − 1 π 1 3 π 1 5 = (1 δ)F ,β , ,ε,δ + δF ,β , ,ε,δ . (26) 2 − 1 2 ǫ 2 6 1 2 ǫ 2 Corollaries of Theorem 1.

A few special cases follow. In the case: a = b = c the two variables of the hypergeometric function of Appell that appear in6 (22) become equal and consequently F1 reduces to the hypergeometric function of Gauß: 1 1 3 3 c2 c2 1 1 3 3 c2 F , , , , 1 , 1 = F , + , , 1 (27) 1 2 −2 2 2 − a2 − a2 2 −2 2 2 − a2 and (22) takes the form:

2 2 ελλειψoειδǫς´ 8a π 1 1 3 c a=b=c = 2 2 3/2 F , 1, , 1 2 A 6 √1 δ 4 [1/(1 δ)] 2 2 − a − ( − π ( δ) 1 1 4 1 5 c2 + − F , 1, , 1 − 4 1 δ [1/(1 δ)]3/2 2 3 2 2 − a2 − − ) (28)

Equation (28) admits a further simpliﬁcation:

Corollary 3 In the special case of an ellipsoid with a = b>c ( oblate spheroid) the following equation is valid:

2 2 ελλειψoειδǫς´ 8a π 1 1 3 c a=b=c = 2 2 3/2 F , 1, , 1 2 A 6 √1 δ 4 [1/(1 δ)] 2 2 − a − ( − π ( δ) 1 1 4 1 5 c2 + − F , 1, , 1 − 4 1 δ [1/(1 δ)]3/2 2 3 2 2 − a2 − − ) 1 1+ 2 1 c2/a2 =2πa2 + πc2 log − . 2 1 c2/a2 1 2 1 c2/a2 − − p − p p (29) Proof. We are going to make use of the formula [14] : 1+ x 1 3 log =2xF , 1, , x2 , (30) 1 x 2 2 − and the contiguous equation: γ d F (α,β,γ+1,z)= (1 z) + γ α β F (α,β,γ,z). (31) (γ α)(γ β) − dz − − − −

7 Equation (30) yields:

2 2 2 2 1 3 c 1 1+ √1 c /a F , 1, , 1 2 = log − , (32) 2 2 − a 2 √2 1 c2/a2 1 √2 1 c2/a2 − − − while (31)

2 2 2 2 2 2 c /a 1+ √1 c /a 1 5 c 3 1 1 − F , 1, , 1 2 = 2 2 2 3/2 log 2 . (33) 2 2 a 2 1 c /a 2 c 2 2 − − 1 2 1 √1 c /a ( − a − − ) − Substituting equations (32), (33) into (28) the corollary is proved.

Corollary 4 In the special case of an ellipsoid with a>b = c (prolate spheroid) it holds:

2 2 ellipsoid 8a π 1 1 3 3 c a=b=c = 2 2 3/2 F , , , 1 2 A 6 √1 δ 4 [1/(1 δ)] 2 2 2 − a − ( − π ( δ) 1 1 4 1 3 5 c2 + − F , , , 1 − 4 1 δ [1/(1 δ)]3/2 2 3 2 2 2 − a2 − − ) 2 2 2 2 2 b (1 b /a ) 2 2 2 =2abπ 2 + − 2 2 arcsin 1 b /a . ra p1 b /a − ! − p Proof. Indeed, for b = c, the ﬁrst variable of the Appell’s functions in the closed form solution of (22) vanishes and the generalized hypergeometric functions in discussion reduce as follows: 1 3 c2 1 3 3 c2 F ,β , , 0, 1 = F , , , 1 , (34) 1 2 ε 2 − a2 2 2 2 − a2 1 5 c2 1 3 5 c2 F ,β , , 0, 1 = F , , , 1 (35) 1 2 ε 2 − a2 2 2 2 − a2 We now apply the formulae :

1 1 3 1 F , , , x = arcsin √2 x, (36) 2 2 2 √2 x

γ β d F (α, β +1,γ +1, x) F (α,β,γ,x)= − x F (α,β,γ +1, x), − βγ dx (37) x d F (α, β +1,γ,x)= F (α,β,γ,x)+ F (α,β,γ,x). β dx (38)

8 We thus end up with the equations:

1 3 3 1 F , , , x = , (39) 2 2 2 √2 1 x − 1 3 5 3 arcsin √2 x 3 √2 1 x F , , , x = − , (40) 2 2 2 2 x3/2 − 2 x and the corollary is proved.

Corollary 5 In the special case of an ellipsoid with a = c

8ab π 1 1 3 3 c2 ellipsoid = F , , , , 1 , 0 Aa=c

F1(α,β,β′, γ, x, 0) = F (α,β,γ,x), (42) 1 d F (α, β 1,γ,x)= z(1 z) αz β + γ F (α,β,γ,x) (43) − γ β − dz − − − and (36).

Corollary 6 In the particular case of the sphere a = b = c (22) reduces to the known result =4πa2. A Let us give some examples of (22). For a =2,b =1,c =0.25 using (22) we compute: ellipsoid = 13.6992108087. For a = 1,b = 1,c = 0.5, we calculate ellipsoid =A 8.6718827033, and for a = 1,b = 0.8,c = 0.625, we compute: Aellipsoid =8.1516189229. A

3 Capacitance of a conducting ellipsoid.

For a conducting 3-dimensional ellipsoid E given by equation (1) its electrostatic capacitance is determined by solving Laplace’s equation:

∂2Φ ∂2Φ ∂2Φ 2Φ= + + =0, (44) ∇ ∂x2 ∂y2 ∂z2

9 outside E, subject to Φ = 1 on the surface and Φ = 0 at . The function Φ(~x) is the (equilibrium) electrostatic potential of E. ∞ The computation is facilitated by ellipsoidal coordinates, a trick ﬁrst intro- duced by Jacobi [15]. A point ~x outside E determines the parameter 0 r = r(~x) < by means of ≤ ∞ x2 y2 z2 + + =1. (45) a2 + r b2 + r c2 + r Now we look at:

1 ∞ 1/2 Φ(~x)= (a2 + r)(b2 + r)(c2 + r) − dr (46) 2 Zr(~x) outside E. It satisﬁes (44) and the prescribed conditions, therefore the capacitance is given by [17],[18]:

1 1 ∞ dr C− (E)= (47) 2 2 (a2 + r)(b2 + r)(c2 + r) Z0 We will now show the following theoremp in which we compute exactly the elliptic integral in (47):

Theorem 7 The closed form solution for the capacitance of a 3-dimensional conducting ellipsoid is given by the expression:

2aΓ(3/2) 1 C(E)= 1 1 1 3 , Γ(1/2)Γ(1) F1 2 , 2 , 2 , 2 ,x,y (48) 2 2 b c where x := 1 2 ,y := 1 2 and we organize the axes so that: a>b>c> 0. − a − a Equivalently, by substituting the values for the gamma function into (48): 2 √π 2 Γ(3/2) = 2 , Γ(1/2) = √π, we derive: a C(E)= 1 1 1 3 b2 c2 . (49) F , , , , 1 2 , 1 2 1 2 2 2 2 − a − a For the proof of theorem 7 see Appendix. We also derive the following corollaries of Theorem 7:

Corollary 8 For the special case a = b>c the capacitance of the spheroid is given by 2aΓ(3/2) 2 1 c2/a2 Ca=b>c = − . (50) Γ(1/2) arcsin 2 1 c2/a2 p − p

10 Proof. For a = b > c, Eq. (49) reduces to:

2aΓ(3/2) 1 C = 1 1 1 3 c2 Γ(1/2) F ( , , , , 0, 1 2 ) 1 2 2 2 2 − a 2 1 (36) 1 c2/a2 = a 2 = a − . (51) 1 1 3 c 2 2 2 F , , , 1 2 arcsin 1 c /a 2 2 2 − a p − p

Corollary 9 For b = c < a, the case of a prolate spheroid we derive

2 2 1 c2/a2 Cb=c

2aΓ(3/2) 1 C = 1 1 1 3 c2 c2 Γ(1/2) F , , , , 1 2 , 1 2 1 2 2 2 2 − a − a 2 1 (30) 2 1 c2/a 2 = a 2 = a − (53) 1 3 c 2 2 2 F , 1, , 1 2 1+ √1 c /a 2 2 a logp 2 − − 1 √1 c2/a2 − −

Corollary 10 For a conducting sphere, a = b = c, and equation (48) reduces to: Csphere = a. (54)

Proof. For the case of the sphere the ﬁrst hypergeometric function of Appell in (48) takes the value 1.

Corollary 11 For the case of an elliptic disk, a>b>c = 0, the capacitance is given in closed analytic form by: a Celliptic disk = π 1 1 b2 . (55) F , , 1, 1 2 2 2 2 − a Proof. Indeed, in the case of an elliptic disk equation (48) reduces as follows: a C = 1 1 1 3 b2 F1 2 , 2 , 2 , 2 , 1 a2 , 1 − a = Γ(3/2)Γ(3/2 1/2 1/2) 1 1 3 1 b2 − − F , , , 1 2 Γ(3/2 1/2)Γ(3/2 1/2) 2 2 2 − 2 − a − a − = π 1 1 b2 (56) F , , 1, 1 2 2 2 2 − a

11 a =5,b =2,c =1 a = 10,b =5,c =4 a = 10,b =8,c =5 C/a =0.50822148949 C/a =0.621383016235 C/a =0.76121621804

Table 1: Capacitance of some ellipsoids computed from (48)

We now apply our closed form analytic solution (49) for computing the capacitance of some ellipsoids. Our results are presented in Table 1.

In Figure 1 , we plot the capacitance C(E)/a using Eqn.(49), of a conducting ellipsoid immersed in R3 versus the ratio c/a of the axes for various values of the ratio b/a.

4 Demagnetizing factors of a magnetized ellipsoid

The first attempts of calculating the demagnetizing factors of the ellipsoid were made in [19],[20]. However, the authors in [19],[20], only derived expressions in terms of formal integrals. We now derive the first closed form analytic solution for the demagnetizing factors of the magnetized ellipsoid in terms of Appell’s first hypergeometric function F1. The potential in the interior of the magnetized ellipsoid is a quadratic function of x, y, and z [19]: Φ = Lx2 My2 Nz2 + W, (57) int − − − where the demagnetizing factors are defined by the integrals:

∞ du L = πa a a κ , (58) 1 2 3 2 2 2 2 0 (a + u) (a + u)(a + u)(a + u) Z 1 1 2 3 ∞ du M = πa a a κ p , (59) 1 2 3 2 2 2 2 0 (a + u) (a + u)(a + u)(a + u) Z 2 1 2 3 ∞ du N = πa a a κ p , (60) 1 2 3 2 2 2 2 0 (a + u) (a + u)(a + u)(a + u) Z 3 1 2 3 where we have the correspondence: p

a1 = a,a2 = b,a3 = c, (61) a > b > c. (62) and ∞ du W = πa a a κ . (63) 1 2 3 2 2 2 0 (a + u)(a + u)(a + u) Z 1 2 3 From Poisson’s diﬀerential equationp 2Φ = 2(L + M + N)= 4πκ, (64) ∇ int − −

12 we derive the relationship that the demagnetizing factors satisfy: L + M + N =2πκ. (65) Theorem 12 The closed form solution of the demagnetizing factors L,M,N, of the magnetized scalene ellipsoid is the following: 3 2 2 πabc Γ 2 Γ(1) 3 1 1 5 b c L = κ F1 , , , , 1 , 1 , (66) a3 Γ 5 2 2 2 2 − a2 − a2 2 3 2 2 πabc Γ 2 Γ(1) 3 3 1 5 b c M = κ F1 , , , , 1 , 1 , (67) a3 Γ 5 2 2 2 2 − a2 − a2 2 3 2 2 πabc Γ 2 Γ(1) 3 1 3 5 b c N = κ F1 , , , , 1 , 1 . (68) a3 Γ 5 2 2 2 2 − a2 − a2 2 Proof. The demagnetizing factors are deﬁned by the integrals eqns(58)-(60). We now compute these integrals in closed analytic form. We apply the transformation: 2 u 1 2a1 1+ 2 = 2 du = 3 dx. (69) a1 x ⇒ − x Consequently: 2πa a a κ 1 dx x2 L = 1 2 3 , (70) 3 2 2 2 a1 0 (1 µ x )(1 µ x ) Z − 2 − 3 where we deﬁned the moduli: p 2 2 a2 a3 µ2 := 1 2 , µ3 := 1 2 . (71) − a1 − a1 We now set: dξ x2 = ξ dx = . (72) ⇒ 2√2 ξ Thus 1 1/2 πa1a2a3κ dξξ L = 3 2 a1 0 (1 µ ξ)(1 µ ξ) Z − 2 − 3 2 2 (25) πabcκ Γ(3/2)Γ(1) 3 1 1 5 b c = p F , , , , 1 , 1 (73) a3 Γ(5/2) 1 2 2 2 2 − a2 − a2 In a similar way, repeating the previous transformations, we compute analytically the two other demagnetizing factors M,N. For instance: 1 1/2 πa1a2a3κ dξξ N = 3 2 a1 0 (1 µ ξ) (1 µ ξ)(1 µ ξ) Z − 3 − 2 − 3 πbcκ Γ(3/2)Γ(1) 3 1 3 5 b2 c2 = F ,p, , , 1 , 1 . (74) a2 Γ(5/2) 1 2 2 2 2 − a2 − a2

We now study special cases for the demagnetizing factors of the magnetized ellipsoid.

13 Corollary 13 In the special case a = b>c

3 2 c Γ 2 Γ(1) 3 1 1 5 c L = π κ F1 , , , , 0, 1 a Γ 5 2 2 2 2 − a2 2 c Γ 3 Γ(1) 3 1 5 c2 πκm2 √2 m2 1 πκ = π κ 2 F , , , 1 = arcsin − , a Γ 5 2 2 2 − a2 (m2 1)3/2 m − m2 1 2 − − (75) c Γ 3 Γ(1) 3 1 5 c2 πκm2 √2 m2 1 πκ M = π κ 2 F , , , 1 = arcsin − , a Γ 5 2 2 2 − a2 (m2 1)3/2 m − m2 1 2 − − (76) 3 2 c Γ 2 Γ(1) 3 1 3 5 c N = π κ F1 , , , , 0, 1 a Γ 5 2 2 2 2 − a2 2 2 2 3 2 2 √m 1 c Γ 2 Γ(1) 3 3 5 c 2πκm arcsin m− = π κ 5 F , , , 1 2 = 2 − 2 +1 a Γ 2 2 2 − a m 1 ( √m2 1 ) 2 − − (77) where m := a/c.

Proof. For a = b > c,

c Γ 3 Γ(1) 3 1 5 c2 L = M = π κ 2 F , , , 1 . (78) a Γ 5 2 2 2 − a2 2 Using the contiguous relations d z F (α,β,γ,z)= α [F (α +1,β,γ,z) F (α,β,γ,z)] , (79) dz − and Eq.(31)

:=z 1 3 1 5 c2 1 1 1 5 d 1 1 5 F  , , , 1  = F , , ,z + z F , , ,z , (80) 2 2 2 2 − a2 2 2 2 2 dz 2 2 2 z }| {       with 1 1 5 3 1 1 1 1 arcsin √2 z F , , ,z = (1 z) arcsin √2 z + + . 2 2 2 2 − −2z3/2 2z √2 1 z 2 √2 z − (81) Consequently:

πκm2 √2 m2 1 πκ L = M = arcsin − , (82) (m2 1)3/2 m − m2 1 − −

14 2 c 1 where z =1 a2 := 1 m2 . On the other hand, using the contiguous relation (79) for the calculation− − of the N-demagnetizing factor, yields:

d z F (α,β,γ,z)= α [F (α +1,β,γ,z) F (α,β,γ,z)] (83) dz − ⇒ 1 3 3 5 1 1 3 5 d 1 3 5 F , , ,z = F , , ,z + z F , , ,z , (84) 2 2 2 2 2 2 2 2 dz 2 2 2 1 3 5 where the Gauß function F 2 , 2 , 2 ,z is given by Equation (40). Consequently,

1 3 3 5 6 arcsin √2 z 3 √2 1 z 3 1 1 F , , ,z = + − + +1 , (85) 2 2 2 2 −4 z3/2 4 z 4 √2 1 z z − and N is given by Equation (77).

Corollary 14 In the special case b = c

πc2 κΓ 3 Γ(1) 3 5 c2 L = 2 F , 1, , 1 a2 Γ 5 2 2 − a2 2 2 2πκ m m + √m2 1 = 2 log 2 − 1 ,m := a/c, (86) m 1 ( 2 m √m2 1 − ) − − − πc2 κΓ 3 Γ(1) 3 5 c2 N = 2 F , 2, , 1 a2 Γ 5 2 2 − a2 2 2 m 1 1 m + √m2 1 = κπ 2 2 log 2 − + m (87) m 1 "−2 √m2 1 m √m2 1 # − − − − Proof. For b = c

2 3 2 2 πac Γ 2 Γ(1) 3 1 3 5 c c N = κ F1 , , , , 1 , 1 a3 Γ 5 2 2 2 2 − a2 − a2 2 z:= πac2 Γ 3 Γ(1) 3 5 c2 πc2 κΓ 3 Γ(1) d 1 3 = κ 2 F  , 2, , 1  = 2 3 F , 1, ,z a3 Γ 5 2 2 − a2 a2 Γ 5 dz 2 2 2 z }| { 2     2 3 2 (30) πc κΓ Γ(1) 3 1 1 1+√z 1 = 2 log + a2 Γ 5 2 −2 z3/2 1 √2 z z(1 z) 2 − − 2 2 z=1 1/m m 1 1 m + √m2 1 − = κπ 2 2 log 2 − + m (88) m 1 "−2 √m2 1 m √m2 1 # − − − −

15 a =3,b =2,c =1 a =4,b =3,c =2 a = 10,b =3,c =2 L L L 4π =0.156300698829271 4π =0.211265605319304 4π =0.0725156494555862 M M M 4π =0.267154040262005 4π =0.305006257867421 4π =0.366221770806668 N N N 4π =0.576545260908724 4π =0.483728136813275 4π =0.561262579737746 Table 2: The demagnetizing factors L/4π,M/4π,N/4π as computed from the equations of Theorem 12, for various ellipsoids.

c b L M N a a 4π 4π 4π Theorem 12 0.0134714 1. 0.017452 0.99620 0.0133953 0.973133 Numerical Osborn 0.013471 Theorem 12 0.0613072 0.062672 0.876021 2. 0.087156 0.984920 Numerical Osborn 0.06108 0.06281 0.87611 Theorem 12 0.235445 0.238955 0.5256 3. 0.5 0.98863 Numerical Osborn 0.23555 0.23885 0.5256 Theorem 12 0.0615658 0.0615658 0.876868 4. 0.087156 1 Numerical Osborn 0.06154 0.06154 0.87692

Table 3: The demagnetizing factors for various values of the ratios c/a, b/a and a comparison with the numerical results of reference [20]. Empty entries in the table means that the author of [20] did not provide such values.

Corollary 15 For a magnetized sphere,a = b = c, and the demagnezing factors are all equal to: 2πκ L = M = N = . (89) 3 Proof. In this case, the generalized hypergeometric functions of Appell that appear in the exact solutions for the demagnetizing factors, eqns(66) (68), take the value 1. −

We now apply our closed form analytic solutions for the demagnetizing factors, Eqns. (66) (68), in order to compute these factors for various ellipsoids. Our results are summarized− in Tables 2 1, 3. We also plot the demagnetizing factors L/4π,M/4π,N/4π of the magnetized ellipsoid versus the ratio c/a, for various values of the ratio b/a. Our results are displayed in Figures 2,3,4.

1Osborn [20] gave the value: M/4π = 0.306 for the particular ellipsoid, second column of Table 2.

16 C HEL

1.0

0.9 dashed curve: a=b; oblate sheroid 0.8

0.8 0.7

0.6

0.5 dashed curve: b=c; prolate spheroid 0.6 0.4

0.3

0.2 0.4

ba: 0.1 c

0.2 0.4 0.6 0.8 1.0 a

0.2

Figure 1: The capacitance C(E) of a conducting ellipsoid immersed in R3 versus the ratio c/a of the axes for various values of the ratio b/a.

17 L 4 Π

0.30 0.9

0.8

0.25 dashed curve: a=b; oblate sheroid 0.7

0.6 0.20

0.5

0.15

0.4 dashed curve: b=c; prolate spheroid

0.10

0.3

0.05 0.2

ba: 0.1 c 0.2 0.4 0.6 0.8 1.0 a

Figure 2: The L demagnetizing factor versus the ratio c/a for various values of the ratio b/a. −

18 M 4 Π 0.5

b : 0.1 a 0.2 0.3 0.4 0.4 dashed curve: b=c; prolate sheroid 0.5 0.6 0.7 0.8 0.9

0.3

dashed curve: a=b; oblate spheroid

0.2

0.1

c 0.2 0.4 0.6 0.8 1.0 a

Figure 3: The M demagnetizing factor versus the ratio c/a for various values of the ratio b/a. The− dashed curves meet at the point determined in Corollary 15.

19 N

4 Π 1.0

0.9

0.8

0.7 dashed curve: a=b; oblate spheroid

0.6

c b 0.2 0.4 0.6 0.8 1.0 a : 0.1 a 0.20.3 0.4 0.4 dashed curve: b=c; prolate sheroid 0.5 0.6 0.7 0.8 0.9

Figure 4: The N demagnetizing factor versus the ratio c/a for various values of the ratio b/a.

20 Using theorem 12 we can write for the potential Φint : x2 y2 z2 Φ = 1 + + , (90) int A − α2 β2 γ2 where 2 2 1 1 1 3 b c F1 , , , , 1 2 , 1 2 2 2 2 2 − a − a 2 α := 3 3 1 1 5 b2 c2 a , (91) F , , , , 1 2 , 1 2 1 2 2 2 2 − a − a 2 2 1 1 1 3 b c F1 , , , , 1 2 , 1 2 2 2 2 2 − a − a 2 β := 3 3 3 1 5 b2 c2 b , (92) F , , , , 1 2 , 1 2 1 2 2 2 2 − a − a 2 2 1 1 1 3 b c F1 , , , , 1 2 , 1 2 2 2 2 2 − a − a 2 γ := 3 3 1 3 5 b2 c2 c , (93) F , , , , 1 2 , 1 2 1 2 2 2 2 − a − a b c 1 1 1 3 b2 c2 := 2κπ a2F , , , , 1 , 1 . (94) A a a 1 2 2 2 2 − a2 − a2 5 Concluding remarks

We have solved analytically a number of problems related to the geometrical and physical properties of the theory of ellipsoid. In particular, we have solved in closed analytic form for the capacitance of a conducting ellipsoid immersed in the Euclidean space R3. The exact solution has been expressed in terms of Appell’s first hypergeometric function F1 and it is given by Theorem 7 and equation (48). We have also computed exactly the capacitance of a conducting ellipsoid in n dimensions. The resulting exact analytic solution is expressed in terms of Lauricella’s− fourth hypergeometric function FD of n 1-variables, see Theorem 16, equation (95). We subsequently− solved analytically for the demagnetizing factors of a magnetized ellipsoid immersed in the Euclidean space R3. The resulting solutions are expressed elegantly in terms of Appell’s first hypergeometric function F1, as stated in Theorem 12 and equations (66) (68). Finally, we have derived the closed form− solution for the geometrical entity of the surface area of the ellipsoid immersed in the Euclidean space R3. Our analytic solution in this case is given in Theorem 1, eqn.(22). We believe that the useful exact analytic theory of the ellipsoid we have developed in this work will have many applications in various scientific fields. We have already outlined in the introduction possible multidisciplinary applications of our theory in science; a scientific multidisciplinarity which measures from physics, biology and chemistry to micromechanics, space science and as- trobiology. A fundamental mathematical generalization of our theory would be to inves- tigate the immersion of an ellipsoid in curved spaces 2and solve for the corre- 2Some initial steps along this direction have been taken in [5],[25].

21 sponding geometrical and physical properties of such an object. However, such a project is beyond the scope of the present paper and it will be a subject of a future publication.

A Appendix

The generalization of Theorem 7 for the capacitance of the ellipsoid in n- dimensions involves the analytic computation of a hyperelliptic integral. Hyper- elliptic integrals which are involved in the solution of timelike and null geodesics in Kerr and Kerr-(anti) de Sitter black hole spacetimes have been computed analytically in references [21],[22], in terms of the multivariable Lauricella’s hypergeometric function FD. The idea is to bring a hyperelliptic integral by the appropriate transformations onto the integral representation that the function 3 FD admits . Applying this method for the analytic computation of the capacitance of the ellipsoid in n-dimensions, generalizes Theorem 7 to the following one:

Theorem 16 The closed form solution for the capacitance of a n dimensional conducting ellipsoid is given by the formula: −

n 2 n 2a1 − Γ 2 1 C = n 2 (95) Γ −2 Γ(1) n 2 1 1 1 n FD  − , , ,..., , , x2, x3,...,xn 2 2 2 2 2  n 1   −    where FD denotes the fourth hypergeometric| {z function} of Lauricella of n 1- variables and −

2 2 2 a2 a3 an x2 := 1 2 , x3 := 1 2 ,...,xn := 1 2 . (96) − a1 − a1 − a1 Proof. Applying the transformation (69) to the hyperelliptic integral

1 1 ∞ du = , (97) 2 2 2 C 2 0 (a + u)(a + u) (a + u) Z 1 2 ··· n 3For an application of the methodp in the realm of number theory see [23].

22 yields:

1 n 3 1 1 x − dx = n 2 2 C a − (1 µ x2)(1 µ x2) (1 µ x2) 1 Z0 2 3 n − −n−3−1 ··· − 2 1 x =ξ 1 p ξ 2 dξ = n 2 2 2a − 0 (1 µ ξ)(1 µ ξ) (1 µ ξ) 1 Z − 2 − 3 ··· − n

n 2p 1 Γ −2 Γ(1) n 2 1 1 1 n = FD  − , , ,..., , , x2, x3,...,xn (98) n 2 n 2 2 2 2 2 2a1 − Γ 2  n 1   −    where | {z } 2 aj µj xj := 1 2 , j =1, 2, . . . , n, (99) ≡ − a1 and a a ... a > 0 (100) 1 ≥ 2 ≥ ≥ n For n = 3 we derive 1 1 Γ(1/2)Γ(1) 1 1 1 3 b2 c2 = F , , , , 1 , 1 , (101) C 2a Γ(3/2) 1 2 2 2 2 − a2 − a2 1 and thus Theorem 7 is proved as well. Applying our closed form analytic formula, eqn.(95), for n = 4, and the choice of values a1 = 2,a2 =1+2/3,a2 =1+1/3,a4 = 1 we derive for the capacitance of this particular higher dimensional ellipsoid:

C(n=4) =4.406592791665676649174487. (102)

The generalization of Theorem 12 in n-dimensions is the following:

Theorem 17

n (n) πa1a2a3 ...an Γ 2 Γ(1) n 1 1 1 n +2 = FD  , , ,..., , ,µ2,...,µn , L an n+2 2 2 2 2 2 1 Γ 2  n 1   −    | {z } (103) n (n) πa1a2a3 ...an Γ 2 Γ(1) n 3 1 1 n +2 = FD , , ,..., , ,µ2,...,µn , M an Γ n+2 2 2 2 2 2 1 2 (104)

The fourth hypergeometric function of Lauricella FD of m-variables [24] is deﬁned as follows:

23 ∞ (m) (α)n1+ nm (β1)n1 (βm)nm n1 nm FD(α, β, γ, z) FD (α, β, γ, z) := ··· ··· z1 zm ≡ (γ)n1+ +n (1)n1 (1)n ··· n1,n2,...,n =0 m m X m ··· ··· (105) where

z = (z1,...,zm),

β = (β1,...,βm). (106)

The Pochhammer symbol (α)m = (α, m) is deﬁned by

Γ(α + m) 1, if m =0 (α) = = (107) m Γ(α) α(α + 1) (α + m 1) if m =1, 2, 3 ··· − The series admits the following integral representation:

1 Γ(γ) α 1 γ α 1 β1 β F (α, β, γ, z)= t − (1 t) − − (1 z t)− (1 z t)− m dt D Γ(α)Γ(γ α) − − 1 ··· − m − Z0 (108) which is valid for Re(α) > 0, Re(γ α) > 0. . It converges absolutely inside − the m-dimensional cuboid

z < 1, (j =1,...,m). (109) | j | It also has the following values:

(n) FD (α, β1,...,βn, γ, 1, x2,...,xn) Γ(γ)Γ(γ α β1) (n 1) = − − F − (α, β ,...,β ,γ β , x ,...,x ), (110) Γ(γ α)Γ(γ β ) D 2 n − 1 2 n − − 1 when max x ,..., x < 1, Re(γ α β ) > 0. {| 2| | n|} − − 1 References

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